The weaker side cannot win the pawn, nor has a perpetual check nor can play on stalemate.
Is there any drawing position left with all these conditions?
I will answer this with a no, and for a reason that's more general than the KQPKQ scenario you're specifically interested in. But ultimately, whether the answer to your question is yes or no comes technically down to definitions. First, consider this KQPKQ position, with White to move:
[FEN "8/5k2/5Pq1/4K3/8/8/8/5Q2 w - - 0 32"]
This is a drawn position, and perhaps one might be tempted to say it meets your criteria: White never needs to lose the pawn, nor allow a perpetual check from Black, nor play into a stalemate. So why don't I think this gives a "yes" answer to your question? Because even so, the only reason it's a draw is because White can only avoid those fates by playing into a different repetition himself.
My more general point is this: there are really only two essential reasons a game is ever drawn (ignoring mutual agreement, since we're talking theory).
Perpetual check is a particular case where one side forces a repetition directly, and the 50-move rule is theoretically redundant: if mate can't be forced, players will unavoidably fall into a repetition once there are no more captures or pawn moves available (as there are only finitely many chess positions in total). The 50-move rule is merely a convenience for practice. Even insufficient mating material as a reason is theoretically redundant, as again, if a mate can't be forced and stalemate doesn't arise, eventually repetition will crop up.
What I'm getting at is, even beyond your KQPKQ endgame, the only base reason a position is ever theoretically drawn is because either stalemate or repetition is ultimately unavoidable; that's it. You could take my example position as giving your question a "yes" answer, since White can choose to enter a repetition that isn't Black perpetually checking the white king; my preference is to say the answer is still "no" as I consider that functionally equivalent to allowing a perpetual from Black, but it's all semantics at that point.
A preemptive point of clarification. It is of course the case that the set of positions which are drawn with the 50-move rule in effect is different than the set of positions which are drawn without it in effect. So the 50-move rule has a definite impact on "practical endgame theory." The sense in which I'm calling it "theoretically redundant" above is only that, in an ideal setting chess never needed such a rule in the first place, as draw by repetition would eventually arise for any position in which progress can't be made.